= Symmetric two-propagator expansion
{title2=$[(p-k)^2-m^2]^{-1}[(p+k)^2-m^2]^{-1}$}
Set $A=p^2-m^2\ne0$. The denominator product is exactly $(A+k^2)^2-4(p\cdot k)^2$. Its reciprocal therefore has the <Taylor expansion>
$$
\frac1{[(p-k)^2-m^2][(p+k)^2-m^2]}
=\frac1{A^2}\left[1-\frac{2k^2}{A}+\frac{4(p\cdot k)^2}{A^2}+O(k^4)\right].
$$
The coefficients follow by expanding $1/(1+x)=1-x+O(x^2)$ with $x=2k^2/A-4(p\cdot k)^2/A^2+O(k^4)$. The expansion is a local algebraic statement away from denominator zeros; integrating it through <Feynman propagator> poles requires a regulator and the correct <Feynman i-epsilon prescription>.
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